Computer implemented system for determining a distribution policy for a single period inventory system, optimization application therefor, and method therefor, and decision support tool for facilitating user determination of a distribution policy for a single period inventory system

ABSTRACT

Computer implemented system for determining a distribution policy for a single period inventory system on the basis of the relative merit of allocating a draw unit of one of a multitude of different consumer items to one of a multitude of different locations in accordance with an allocation decision criterion subject to one or more constraints, an optimization application therefor, and method therefor, and including a Decision Support Tool for facilitating user determination of a distribution policy for a single period inventory system.

FIELD OF THE INVENTION

The invention is in the field of determining distribution policies forsingle period inventory systems.

GLOSSARY OF TERMS

The following terms listed alphabetically together with their acronymsare employed in the description and claims of this application withrespect to the present invention:

Availability A_(ij), and System Availability Percentage % SA

Availability A_(ij) is an industry term referring to the probability ofcompletely satisfying the demand for an i^(th) consumer item where i=1,2, . . . , m at a j^(th) location where j=1, 2, . . . , n of a singleperiod inventory system without an occurrence of a sellout due toinsufficient draw at that location. In mathematical terms,A_(ij)=F(λ_(ij),D_(ij)) where F is the cumulative probabilitydistribution function (cdf) of demand for the i^(th) consumer item atthe j^(th) location, and λ_(ij) and D_(ij) are its mean demand and draw,respectively. The probability distribution function in the discrete caseand the probability density function in the continuous case are bothdenoted by the letter “f”. The System Availability Percentage % SA for asingle period inventory system is given by % SA=100ΣΣA_(ij)/mn=100−%ESO.Demand X_(ij), Mean Demand λ_(i), and Mean Demand Matrix λThe demand process for a consumer item at a location has a random butnon-stationary nature, and therefore cannot be subjected to ensembleinferences based on a single realization. Mean demands λ_(ij) for aconsumer item at a location over time are presupposed to be the outcomeof a stochastic process which can be simulated by a forecast modelwhilst the demand X_(ij) for an i^(th) consumer item at a j^(th)location of a single period inventory system at a future point in timeis a random variable with a conditional probability distributionconditioned on its mean demand λ_(ij) at that point in time. A meandemand matrix λ is a matrix of mean demands λ_(ij).Distribution PolicyA distribution policy is the delivered quantities of each i^(th)consumer item where i=1, 2, . . . , m at each j^(th) location where j=1,2, . . . , n of a single period inventory system in accordance with apredetermined business strategy.Draw D_(ij), and Draw Matrix DDraw D_(ij) is an industry term referring to the delivered quantity ofan i^(th) consumer item to a j^(th) location of a single periodinventory system. A draw matrix D is a matrix of draws D_(ij).Returns R(λ_(ij),D_(ij)), Total Expected Returns ER(λ,D), and ExpectedReturns Percentage % ERReturns R(λ_(ij),D_(ij)) is an industry term referring to the number ofunsold items of an i^(th) consumer item at a j^(th) location of a singleperiod inventory system, and is given byR(λ_(ij),D_(ij))=max(D_(ij)−X_(ij),0) where D_(ij), X_(ij), and λ_(ij)are the i^(th) consumer item's draw, demand and mean demand,respectively, at the j^(th) location. The total expected returns ER atall n locations of a single period inventory system is given byER(λ,D)=ΣΣER(λ_(ij), D_(ij)) where ER(λ_(ij),D_(ij)) is the expectedvalue of R(λ_(ij),D_(ij)). The expected returns percentage % ER of adistribution policy for a single period inventory system is given by %ER(λ,D)=100ER(λ,D)/ΣΣD_(ij)=100−% ES(λ,D).Safety Stock SS_(i) and Total Safety Stock QFor the purpose of the present invention, safety stock SS_(i) refers tothe difference between an actual draw of an i^(th) consumer item at aj^(th) location of a single period inventory system and its demandforecast at that location, namely, SS_(ij)=D_(ij)−λ_(ij), and thereforecan assume positive or negative values. This is in contradistinction tothe traditional industry definition of safety stock, namely,SS_(ij)=max{0, D_(ij)−λ_(ij)}. The total safety stock Q of all mconsumer items at all n locations of a single period inventory system isgiven by Q=ΣΣSS_(ij).Sales S(λ_(ij),D_(ij)), Total Expected Sales ES(λ,D), and Expected SalesPercentage % ESSales S(λ_(ij),D_(ij)) refers to the number of sold items of an i^(th)consumer item at a j^(th) location of a single period inventory systemas upper bounded by the draw D_(ij) at that location for that consumeritem at each point in time, and is given byS(λ_(ij),D_(ij))=min(D_(ij),X_(ij))=D_(ij)−R(λ_(ij),D_(ij)) whereD_(ij), X_(ij), and λ_(ij) are the i^(th) consumer item's draw, demand,and mean demand, respectively, at the j^(th) location. The totalexpected sales ES(λ,D) of all m consumer items at all n locations of asingle period inventory system is given by ES(λ,D)=ΣΣES(λ_(ij),D_(ij))where ES(λ_(ij),D_(ij)) is the expected value of S(λ_(ij),D_(ij)). Theexpected sales percentage % ES of a distribution policy for a singleperiod inventory system is given by % ES(λ,D)=100ES(λ,D)/ΣΣD_(ij)=100−%ER(λ,D).Sellout SO(λ_(ij),D_(ij)), Expected Number of SelloutsESO(λ_(ij),D_(ij)), Total Expected Number of Sellouts ESO(λ,D), andExpected Sellout Percentage % ESOSellout SO(λ_(ij),D_(ij)) is an industry term referring to an occurrenceof demand being greater than a delivered quantity of an i^(th) consumeritem at a j^(th) location of a single period inventory system, namely,SO(λ_(ij),D_(ij))=δ(D_(ij)<X_(ij)) where δ is a binary indicatorfunction:

${\delta({condition})} = \left\{ \begin{matrix}{1,\;{{if}\mspace{14mu}{condition}\mspace{14mu}\text{is}\mspace{11mu}{true}}} \\{{0,{else}}\mspace{166mu}}\end{matrix} \right.$where D_(ij), X_(ij), and λ_(ij) are the i^(th) consumer item's draw,demand, and mean demand, respectively, at that the j^(th) location. Theexpected number of sellouts ESO(λ_(ij),D_(ij)) for an i^(th) consumeritem at a j^(th) location of a single period inventory system is givenby ESO(λ_(ij),D_(ij))=P(X_(ij)>D_(ij))=1−F(λ_(ij),D_(ij)). The totalexpected number of sellouts of all m consumer items at all n locationsof a single period inventory system is given byESO(λ,D)=ΣΣESO(λ_(ij),D_(ij))=mn−ΣΣF(λ_(ij),D_(ij)). The expectedsellout percentage (% ESO) of a distribution policy for a single periodinventory system is given by % ESO(λ,D)=100ESO(λ, D)/mn=100−% SA.Single Period Inventory SystemsSingle period inventory systems are largely concerned with consumeritems having a limited shelf life at the end of which an item losesmost, if not all, of its consumer value, and the stock of which is notreplenished to prevent an occurrence of a sellout. Such consumer itemscan include perishable goods, for example, fruit, vegetables, flowers,and the like, and fixed lifetime goods, for example, printed mediapublications, namely, daily newspapers, weeklies, monthlies, and thelike. Two common problems of single period inventory systems are knownin the industry as the so-called “newsvendor” problem i.e. the sale ofthe same item throughout a multi-location single period inventory systemand the so-called “knapsack” problem i.e. the sale of different items atthe same location.Stockout ST(λ_(ij),D_(ij)), Expected Stockout EST(λ_(ij),D_(ij)), TotalExpected Stockout EST(λ,D), and Expected Stockout Percentage % ESTStockout ST(λ_(ij),D_(ij)) is the quantity of unsatisfied demand for ani^(th) consumer item at a j^(th) location of a single period inventorysystem, and is given by ST(λ_(ij),D_(ij))=max(X_(ij)−D_(ij),0)=X_(ij)−S(λ_(ij),D_(ij)) where D_(ij), X_(ij) and λ_(ij) are thei^(th) consumer item's draw, demand, and mean demand, respectively, atthe j^(th) location. The total expected stockout EST(λ,D) of all mconsumer items at all n locations of a single period inventory system isgiven by EST(λ,D)=ΣΣEST(λ_(ij),D_(ij)) where EST(λ_(ij),D_(ij)) is theexpected value of ST(λ_(ij),D_(ij)). The expected stockout percentage %EST for a distribution policy is given by %EST(λ,D)=100EST(λ,D)/ΣΣD_(ij).

BACKGROUND OF THE INVENTION

One computer implemented approach for calculating a demand forecastinvolves defining a so-called demand forecast tree capable of beinggraphically represented by a single top level node with at least twobranches directly emanating therefrom, each branch having at least onefurther node. The demand forecast is computed on the basis of historicalsales data typically associated with bottom level nodes of a demandforecast tree by a forecast engine capable of determining a mathematicalsimulation model for a demand process. One such forecast engineemploying statistical seasonal causal time series models of count datais commercially available from Demantra Ltd, Israel, under the nameDemantra™ Demand Planner.

One exemplary demand forecast application is the media distributionproblem, namely, determining the number of copies of different dailynewspapers to be delivered daily to different locations to minimize twomutually conflicting indices commonly quantified for evaluating theefficacy of a distribution policy for a newspaper: the frequency ofsellouts, and the number of returns both typically expressed inpercentage terms. It is common practice in the industry that a draw fora newspaper at a location for a given day is greater than its demandforecast at that location for that day so as to reduce the probabilityof a sellout but with the inherent penalty that returns will be greater.In the case of distribution policies for newspapers, safety stocks areallocated to locations to ensure a predetermined availability level fora given demand probability function to achieve a reasonable balancebetween expected returns and expected occurrences of sellouts. Moreover,it is common practice that locations are sorted into one of severalclasses depending on the average number of copies sold, each class beingassigned a different availability level, say, 70%, 80%, and the like.

SUMMARY OF THE INVENTION

Broadly speaking, the present invention provides a novel computerimplemented system for determining a distribution policy for a singleperiod inventory system on the basis of performance metrics, forexample, returns, sellout, and stockout other than the hitherto employedavailability metric. In contradistinction to prevailing distributionpolicy practice which effectively regards each location of a singleperiod inventory system as an isolated entity, the present invention isbased on the notion that a distribution policy should allocate drawunits on the basis of relative merit in accordance with an allocationdecision criterion subject to one or more constraints rather than insome arbitrary absolute fashion. The choice of the most appropriateallocation decision criterion coupled with one or more constraints for asingle period inventory system is highly dependent on characteristics ofthe single period inventory system in question, for example, thefrequency distribution of the mean demands at its nodes, amongst others,and a business objective.

The preferred allocation decision criteria of the present invention canbe divided into two groups as follows:

Group I consists of simple allocation decision criteria including interalia:

(i) maximum incremental availability max_(i,j){F(λ_(ij),D_(ij)+1)−F(λ_(ij),D_(ij))};

(ii) minimum availability min_(i,j){F(λ_(ij),D_(ij))};

(iii) minimum incremental expected return min_(i,j){ER(λ_(ij),D_(ij)+1)−ER(λ_(ij),D_(ij))}; and

(iv) maximum decremental expected stockout max_(i,j){EST(λ_(ij),D_(ij))−EST(λ_(ij),D_(ij)+1)};

each being subject to one or more of the following constraintsΣΣSS_(ij)≦Q where Q is the total safety stock threshold for delivery ofall m consumer items to all n locations, Σ_(j)SS_(ij)≦q₁ ^(i) where q₁^(i) is the safety stock of the i^(th) consumer item at all locations,Σ_(i)SS_(ij)≦q₂ ^(j) where q₂ ^(j) is the safety stock of all theconsumer items at a j^(th) location % EST(λ,D)≦s where s is apredetermined expected stockout percentage threshold, % ER(λ,D)≦r wherer is a predetermined expected return percentage threshold, % ESO(λ,D)≦ewhere e is a predetermined expected sellout percentage threshold,a_(ij)≦D_(ij)≦b_(ij) where a_(ij) and b_(ij) are respectively lower andupper boundaries for a draw of an i^(th) consumer item at a j^(th)location of a single period inventory system; A≦ΣΣD_(ij)≦B where A and Bare respectively lower and upper boundaries for the draw of all mconsumer items at all n locations of a single period inventory system,A₁ ^(j)≦Σ_(i)D_(ij)≦B₁ ^(j) where A₁ ^(j) and B₁ ^(j) are respectivelylower and upper boundaries for the draw of all m consumer items at aj^(th) location of a single period inventory system, and A₁^(i)≦Σ_(j)D_(ij)≦B₁ ^(i) where A₁ ^(i) and B₁ ^(i) are respectivelylower and upper boundaries for the draw of a i^(th) consumer items atall n locations of a single period inventory system.

Group II consists of weighted composite allocation decision criteriaeach having two components oppositely acting upon the draw matrix Drequired to yield a predetermined business objective expressed in termsof an expected returns percentage (% ER) or an expected percentage of aparameter associated with occurrences of sellouts of all m consumeritems at all n locations of a single period inventory system. Theparameter associated with occurrences of sellouts may be either thenumber of sellouts of all m consumer items at all n locations of asingle period inventory system in which case the allocation decisioncriterion is as follows:

(v) w₁(% ER(λ,D)−% ER(λ,D⁰))+w₂(% ESO(λ,D⁰)−% ESOλ,D)) or

the number of stockouts at all n locations of a single period inventorysystem in which case the allocation decision criterion is as follows:

(vi) w₁(% ER(λ,D)−% ER(λ,D⁰))+w₂(% EST(λ,D⁰)−% EST(λ,D))

where w₁ and w₂ are weights, and D⁰ is an initial draw matrix. Theweighted composite allocation decision criteria can be subject to one ormore of the above mentioned constraints, and also % ER(λ,D)=% ESO(λ,D)in the case of criterion (v), and also % ER(λ,D)=% EST(λ,D) in the caseof criterion (vi). Typically D⁰=λ. In point of fact, the lattercriterion is conceptually more valid than the former criterion since thetwo parameters “returns” and “stockouts” have the same dimensions,namely, units of consumer items, which is not the dimension of sellouts.But this notwithstanding, it is envisaged that the former selloutcriterion will gain more acceptance than the latter stockout criterionsince expected sellout percentages rather than expected stockoutpercentages are more traditional in the art of single period inventorysystems.

To reach an optimal allocation of draw units, the simplest approach isto allocate additional draw units one by one starting from an initialdraw allocation, say, equal to the mean demand matrix. But in the caseof allocating a predetermined total draw quantity ΣΣD_(ij) or totalpredetermined safety stock quantity Q, it may be allocated with lessiterations if it is initially allocated between the locations of asingle period inventory system, say, in accordance with a predeterminedavailability at each location, and thereafter the initial drawallocation is fine-tuned to optimal allocations at each location inaccordance with a selected allocation decision criterion by so-calledpairwise switching.

In connection with the weighted composite allocation decision criteria(v) and (vi), the present invention also provides a computer implementedDecision Support Tool for graphically displaying the expected returnspercentages % ER for a multitude of expected returns percentages againsttheir corresponding minimal expected sellout percentages % ESO, or viceversa. Alternatively, the Decision Support Tool can preferablygraphically display expected returns percentages % ER for a multitude ofexpected returns percentages against their corresponding minimalexpected stockout percentages % EST, or vice versa.

BRIEF DESCRIPTION OF DRAWINGS

In order to better understand the invention and to see how it can becarried out in practice, preferred embodiments will now be described, byway of non-limiting examples only, with reference to the accompanyingdrawings in which:

FIG. 1 is a pictorial representation showing a demand forecast tree forcomputing demand forecast information for five different perishableconsumer items;

FIG. 2 is a table showing historical sales data associated with thedemand forecast tree of FIG. 1;

FIG. 3 is a block diagram of a computer implemented system fordetermining a distribution policy for a single period inventory system,and including a Decision Support Tool for facilitating userdetermination of a distribution policy for a single period inventorysystem;

FIG. 4 is a pictorial representation of a simple single period inventorysystem having three locations for draw allocation in accordance with thepresent invention;

FIG. 5 is a flow chart of a method for determining a distribution policyfor a single period inventory system in accordance with the presentinvention;

FIG. 6 is a flow chart showing the steps of a method for re-allocating apredetermined draw to the locations of a single period inventory systembased on maximal incremental availability in accordance with a firstpreferred embodiment of the method of FIG. 5;

FIG. 7 is a table summarizing the results of the iterations forre-allocating the combined total draw of the demand forecast and 15safety stock units between the locations of the single period inventorysystem of FIG. 4 in accordance with the method of FIG. 6;

FIG. 8 is a flow chart similar to the flow chart of FIG. 6 but for theone-by-one allocation of a predetermined draw to the locations of asingle period inventory system in accordance with a second preferredembodiment of the method of FIG. 5;

FIG. 9 is a table similar to the table of FIG. 7 except in accordancewith the method of FIG. 8 for the one-by-one allocation of up to 20safety stock units;

FIG. 10 is a flow chart showing the steps of a method in accordance withthe present invention for determining a distribution policy for a singleperiod inventory system using a weighted composite allocation decisioncriterion;

FIG. 11 is a table summarizing the minimal expected sellout percentages(% ESO) for a multitude of expected returns percentages (% ER) forallocating draw units to the locations of the single period inventorysystem of FIG. 4 in accordance with the method of FIG. 10 together withtheir corresponding draw vectors D; and

FIG. 12 is a graph showing the results of the table of FIG. 11 forfacilitating user determination of the distribution policy for a singleperiod inventory system.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

FIG. 1 shows an exemplary demand forecast tree 1 having a single toplevel node (00) with five branches A, B, C, D and E for correspondinglyrepresenting the sale of Item I (top level-1 node (10)) at Locations 1and 2 (bottom level nodes (11) and (21)), Item II (top level-1 node(20)) at Locations 1 and 3 (bottom level nodes (21) and (23)), Item III(top level-1 node (30)) at Location 1, 2 and 3 (bottom level nodes (31),(32) and (33)), Item IV (top level-1 node (40)) also at Locations 1, 2and 3 (bottom level nodes (41), (42) and (43)); and Item V (top level-1node (50)) at Location 1 (bottom level node (51)) only. FIG. 2 shows anexemplary table 2 containing historical sales data for Item I at thebottom level nodes (11) and (12). Similar tables exist for the sale ofthe other items at their respective locations.

FIG. 3 shows a computer implemented system 3 with a processor 4, memory6, a user interface 7 including suitable input devices, for example, akeypad, a mouse, and the like, and output means, for example, a screen,a printer, and the like, with other computer components for enablingoperation of the system including result analysis. The computerimplemented system 3 includes a database 8 for storing historical timeseries of sales information of items at locations, a forecast engine 9for forecasting the mean demand λ_(ij) for each i^(th) perishableconsumer item at each j^(th) location on the basis of the historicalsales data, and an optimization application 11 for determining thedistribution policy for a single period inventory system subject to oneor more constraints. The computer implemented system 3 also includes aDecision Support Tool (DST) 12 for facilitating user determination of adistribution policy for a single period inventory system. The computerimplemented system 3 may be implemented as illustrated and described incommonly assigned co-pending U.S. patent application Ser. No. 10/058,830entitled “Computer Implemented Method and System for Demand ForecastApplications”, the contents are which are incorporated herein byreference. Whilst the present invention is being described in thecontext of a fully functional computer implemented system, it is capableof being distributed in as a program product in a variety of forms, andthe present invention applies equally regardless of the particular typeof signal bearing media used to carry out distribution. Examples of suchmedia include recordable type media e.g. CD-ROM and transmission typemedia e.g. digital communication links.

The present invention will now be exemplified for an exemplary“newsvendor” problem for determining the distribution policy for asingle period inventory system 13 for delivering a single newspapertitle between three locations, namely, j=1, 2 and 3 (see FIG. 4). Forthe sake of the example below, demand at Locations 1, 2 and 3 areassumed to have a Poisson probability distribution, and the singleperiod inventory system has a mean demand vector λ=(10, 40, 100) but thepresent invention can be equally applied to other probabilitydistributions of demand. Based on this assumption, the expressions forcalculating expected return (ER) and expected stockout (EST) forLocations 1, 2, and 3 are as follows:ER(λ_(j) ,D _(j))=D _(j) f(λ_(j) ,D _(j)−1)+(D _(j)−λ_(j))F(λ_(j) ,D_(j)−2)EST(λ_(j) ,D _(j))=D _(j) f(λ_(j) ,D _(j))+(λ_(j) −D _(j))(1−F(λ_(j) ,D_(j)−1))where f(•) is the Poisson probability distribution function (pdf) andF(•) is the Poisson cumulative probability distribution function (cdf)for the demand for the consumer item at the j^(th) location, and λ_(j)and D_(j) are respectively the mean demand value and the draw at thatlocation.

The use of the present invention for determining the distribution policyfor the single period inventory system 13 is now described withreference to FIGS. 6-9 in connection with the first simple allocationdecision criterion, namely, maximum incremental availability as given bymax_(j) {F(λ_(j),D_(j)+1)−F(λ_(j),D_(j))} subject to one or more of thefollowing constraints: ΣSS_(j)≦Q, % ER(λ,D)≦r; % ESO(λ,D)≦e; %EST(λ,D)≦s; a_(j)≦D_(j)≦b_(j); and A≦ΣD_(j)≦B. The use of the presentinvention as exemplified in FIGS. 6-10 can be equally extended to theother simple allocation decision criterion (ii) to (iv) by substitutionof their corresponding expressions into the blocks entitled Criterionand Objective in the flow diagrams of FIGS. 6 and 8.

To better exemplify the potential of the present invention for moreadvantageously allocating draw, the following performance metrics % SA,% ESO, % ER, % ES, and % EST are employed for comparing the allocationof the same safety stock quantity in accordance with the conventionalapproach of the same availability at each location and maximumincremental availability. In accordance with a conventional 80%availability at each location, this imposes a safety stock allocation of2, 5 and 8 units to Locations 1, 2 and 3, respectively, which yields thefollowing results: % SA=80.2%, % ESO=19.8%, % ER=10.7%, % ES=89.3%, and% EST=1.6%. FIG. 7 shows how the same safety stock allocation of Q=15units using pairwise switching can arrive at a safety stock allocationof 5, 6 and 4 units to Locations 1, 2 and 3, respectively, which yieldsthe following results: % SA=82.6%, % ESO=17.4%, % ER=10.9%, % ES=89.1%,and % EST=1.8%, namely, a dramatically increased % SA from 80.2% to82.6% whilst paying only marginal penalties in terms of increasedexpected returns percentage (% ER) from 10.7% to 10.9% and increasedexpected stockout percentage (% EST) from 1.6% to 1.8%. The resultsexemplify that since locations with different mean demands contributedifferently to overall system availability % SA, this performance metriccan be improved substantially by allocating draw units to the locationswhich contribute most at a prevailing draw allocation D^(c) at theexpense of other locations. In the present case, the initial safetystock allocation of providing an about 80% availability at each of theLocations 1, 2 and 3 is morphed to availabilities of 95%, 85%, and 68%,respectively.

As mentioned earlier, pairwise switching can only be employed in thecase of re-allocation of a predetermined draw. The table of FIG. 9 showsthe incremental effect of one-by-one allocation of safety stock units tothe Locations 1, 2 and 3, the column entitled “winning location”indicating which Location 1, 2 or 3 receives the next additional safetystock unit on the basis of its incremental availability being thegreatest at any given prevailing draw allocation D^(c). The table ofFIG. 9 enables determining the results of the performance metrics % SA,% ESO, % ER, % ES, and % EST for termination conditions other than apredetermined safety stock quantity, say, % ESO≦15% which in this caseimposes a safety stock allocation of 5, 7 and 6 units to the Locations1, 2 and 3, respectively, which yields % SA=75.9%, % ESO=14.1%, %ER=12.1%, % ES=87.9%, and % EST=1.4%

The use of the present invention for allocating draw to the Locations 1,2 and 3 is now described with reference to FIGS. 10-12 in connectionwith the first weighted composite allocation decision criterion, namely,w₁(% ER(λ,D)−% ER(λ,D⁰))+w₂(% ESO(λ,D⁰)−% ESO(λ,D)) subject to one ormore of the following constraints: ΣSS_(i)≦Q, % ER(λ,D)≦r; % ESO(λ,D)≦e;% EST(λ,D)≦s; a_(i)≦D_(i)≦b_(i); and A≦ΣD_(i)≦B. FIG. 11 shows theresults for repetitions of the method set out in the flow diagram ofFIG. 10 for different expected returns percentage constraints %ER(λ,D)≦r at intervals of about 2% to calculate their correspondingminimal expected sellout percentages % ESO. The DST 12 graphically showsthese results (see FIG. 12) for enabling a user to select a draw vectorD, thereby determining the draw allocation between the Locations 1, 2and 3. This approach can be repeated for a multitude of differentexpected sellout percentage constraints % ESO(λ,D)≦e, say, at intervalsof 5%. Also, this approach may be repeated using the second weightedcomposite allocation decision criterion based on stockouts rather thansellouts.

While the invention has been described with respect to a limited numberof embodiments, it will be appreciated that many variations,modifications, and other applications of the invention can be madewithin the scope of the appended claims.

1. A system for determining a distribution policy for a single periodinventory system, the single period inventory system controlling adistribution of consumer items with limited shelf lives wherein at anend of the limited shelf life the consumer items lose substantially allof their initial consumer value, the system comprising: (a) a databasefor storing historical sales data of sales information of each i^(th)consumer item where i=1, 2, . . . , m at each j^(th) location where j=1,2, . . . , n of the single period inventory system, the database storedin a memory and used by a processor; (b) a forecast engine forforecasting the mean demand λ_(ij) of each i^(th) consumer item at eachj^(th) location of the single period inventory system on the basis ofthe historical sales data; and (c) an optimization applicationcomprising a software module executing on a computer, the softwaremodule operable to: initially allocate draw units to each j^(th)location equal to the mean demand λ_(ij) of an i^(th) consumer item ateach j^(th) location; determine a first location that meets apre-selected allocation decision criterion comprising at least the drawunits allocated to each j^(th) location for the i^(th) customer item,and the mean demand λ_(ij) for the i^(th) customer item at each j^(th)location; allocate a first one and only one additional draw unit to thefirst location; determine a second location that meets the pre-selectedallocation decision criterion with the first location having the firstone and only one additional draw unit, and allocate a second one andonly one additional draw unit to the second location; and determine thesecond location step above until at least one constraint is satisfied,the at least one constraint being based on performance metrics includingat least one of safety stock, expected sellout, expected stockout, orexpected return.
 2. The system according to claim 1 wherein thepre-selected allocation decision criterion is a maximum incrementalavailability, where availability is a probability of completelysatisfying a demand for a first consumer item at a j^(th) locationwithout occurrence of a sellout due to an insufficient draw at thatj^(th) location and is computed using a cumulative distribution function(cdf) of the mean demand λ_(1j) and draw D_(1j) allocated for thatj^(th) location, wherein incremental availability for that j^(th)location is difference between a first availability of the firstconsumer item at that j^(th) location and a second availability of thefirst consumer item at that j^(th) location, the second availabilitycomputed with one additional draw unit D_(1j) of the first consumer itemallocated at that j^(th) location in comparison to the draw unitsallocated for computation of the first availability and the maximumincremental availability is a maximum of the incremental availabilitycomputed for the first consumer item for each j^(th) location, andwherein the determine the first location determines a location that hasthe maximum incremental availability.
 3. The system according to claim 1wherein the pre-selected allocation decision criterion is a minimumavailability, where availability is a probability of completelysatisfying a demand for a first consumer item at a j^(th) locationwithout occurrence of a sellout due to insufficient draw at that j^(th)location and is computed using a cumulative distribution function (cdf)of the mean demand λ_(1j) and draw D_(1j) allocated for the firstconsumer item at each j^(th) location, and wherein the determine thefirst location determines a location that gives a minimum probability.4. The system according to claim 1 wherein the pre-selected allocationdecision criterion is a minimum incremental expected return, whereexpected return is expected number of unsold first consumer item at aj^(th) location of the n locations and is computed using demand X_(1j),mean demand λ_(1j), and draw D_(1j) allocated for the first consumeritem at that j^(th) location wherein, the incremental expected return isdifference between a first expected return and a second expected return,the second expected return computed for one additional draw unit of thefirst consumer item allocated at each j^(th) location in comparison tothe draw units allocated for computation of the first expected returnand wherein the determine the first location determines a location thathas the minimum incremental expected return.
 5. The system according toclaim 1 wherein the pre-selected allocation decision criterion is amaximum decremental expected stockout, where expected stockout isexpected quantity of unsatisfied demand of a first consumer item at aj^(th) location and is computed using demand X_(1j), mean demand λ_(1j)and draw D_(1j) allocated for the first consumer item at that j^(th)location wherein, the incremental expected stockout is differencebetween a first expected stockout and a second expected stockout, thefirst expected stockout computed for one additional draw unit of thefirst consumer item allocated at that j^(th) location in comparison tothe draw units allocated for computation of the second expected stockoutand wherein the determine the first location determines a location thathas the maximum incremental expected return.
 6. The system according toclaim 1 wherein the pre-selected allocation decision criterion is givenby: w₁(% ER(λ, D)−% ER(λ,D⁰))+w₂(% ESO(λ,D⁰)−% ESO(λ,D)) subject to oneor more of the following constraints: ΣΣSS_(ij)≦Q, Σ_(j)SS_(ij)≦q₁ ^(i),Σ_(i)SS_(ij)≦q₂ ^(j), % EST(λ,D)≦s, % ER(λ,D)≦r, % ESO(λ,D)≦e,a_(ij)≦D_(ij)≦b_(ij); A≦ΣΣD_(ij)≦B, A₁ ^(j)≦Σ_(i)D_(ij)≦B₁ ^(j), A₁^(i)≦Σ_(j)D_(ij)≦B₁ ^(i) and % ER(λ,D)=% ESO(λ,D), where w₁ and w₂ areweights, D⁰ is an initial draw matrix, D_(ij) is the draw or deliveredquantity of an i^(th) consumer item to a j^(th) location, SS_(ij) is thesafety stock or difference between the draw of an i^(th) consumer itemat a j^(th) location and a demand forecast at that location, Q is atotal safety stock threshold for delivery of all m consumer items to alln locations, q₁ ^(i) is a safety stock of an i^(th) consumer item at alllocations, q₂ ^(j) is a safety stock of all consumer items at a j^(th)location, EST is expected stockout, ER is expected return, ESO isexpected sellout, s is a predetermined expected stockout percentagethreshold, r is a predetermined expected return percentage threshold, eis a predetermined expected sellout percentage threshold, a_(ij) andb_(ij) are respectively lower and upper boundaries for a draw of ani^(th) consumer item at a j^(th) location, A₁ ^(j) and B₁ ^(j) arerespectively lower and upper boundaries for the draw of all m consumeritems at a j^(th) location, A₁ ^(i) and B₁ ^(i) are respectively lowerand upper boundaries for the draw of an i^(th) consumer item at all nlocations and A and B are respectively lower and upper boundaries forthe draw of all m consumer items at all n locations.
 7. The systemaccording to claim 1 wherein the pre-selected allocation decisioncriterion is oven by: w₁(% ER(λ,D)−% ER(λ,D⁰))+w₂(% EST(λ,D⁰)−%EST(λ,D)) subject to one or more of the following constraints:ΣΣSS_(ij)≦Q, Σ_(j)SS_(ij)≦q₁ ^(i), Σ_(i)SS_(ij)≦q₂ ^(j), % EST(λ,D)≦s, %ER(λ,D)≦r, % ESO(λ,D)≦e, a_(ij)≦D_(ij)≦b_(ij); A≦ΣΣD_(ij)≦B, A₁^(j)≦Σ_(i)D_(ij)≦B₁ ^(j), A₁ ^(i)≦Σ_(j)D_(ij)≦B₁ ^(i) and % ER(λ,D)=%ESO(λ,D), where w₁ and w₂ are weights, D⁰ is an initial draw matrix,D_(ij) is the draw or delivered quantity of an i^(th) consumer item to aj^(th) location, SS_(ij) is the safety stock or difference between thedraw of an i^(th) consumer item at a j^(th) location and a demandforecast at that location, Q is a total safety stock threshold fordelivery of all m consumer items to all n locations, q₁ ^(i) is a safetystock of an i^(th) consumer item at all locations, q₂ ^(j) is a safetystock of all consumer items at a j^(th) location, EST is expectedstockout, ER is expected return, ESO is expected sellout, s is apredetermined expected stockout percentage threshold, r is apredetermined expected return percentage threshold, e is a predeterminedexpected sellout percentage threshold, a_(ij) and b_(ij) arerespectively lower and upper boundaries for a draw of an i^(th) consumeritem at a j^(th) location, A₁ ^(j) and B₁ ^(j) are respectively lowerand upper boundaries for the draw of all m consumer items at a j^(th)location, A₁ ^(i) and B₁ ^(i) are respectively lower and upperboundaries for the draw of an i^(th) consumer item at all n locationsand A and B are respectively lower and upper boundaries for the draw ofm consumer items at all n locations.
 8. The system according to claim 1wherein the software module operable to determine the first locationthat meets the preselected allocation decision criterion is furtheroperable to determine an incremental availability and decrementalavailability for each j^(th) location, select a third location with amaximum incremental availability and a fourth location with a minimumdecremental availability and allocate the one additional draw unit fromthe fourth location to the third location.
 9. The system according toclaim 1 wherein the consumer item is a printed media publication. 10.The system according to claim 1 wherein the pre-selected allocationdecision criterion is a maximum incremental availability criterion andthe determine the first location that meets the pre-selected allocationdecision criterion comprises determines an incremental availability foreach j^(th) location, and selects a location with the maximumincremental availability.
 11. The system according to claim 1 wherein,the safety stock is the difference between an actual draw of the i^(th)consumer item at the j^(th) location and its demand forecast at thatlocation and can assume one of positive, or negative value and whereinthe optimization application is capable of computing the draw units tobe allocated to each jth location even when the safety stock assumes anegative value.
 12. The system according to claim 11 wherein, a negativesafety stock does not cause a violation in the computer-implementedsystem.
 13. A computer-implemented method for determining a distributionpolicy for a single period inventory system, the single period inventorysystem controlling a distribution of consumer items with limited shelflives that will not be replenished to prevent an occurrence of sellout,the method comprising the steps of: (a) storing in a database historicalsales data of sales information of each ith consumer item where i=1, 2,. . . , m at each jth location where j=1, 2, . . . , n of the singleperiod inventory system; (b) forecasting the mean demand λij of each itconsumer item at each jth location of the single period inventory systemon the basis of the historical sales data; (c) initially allocating drawunits to each jth location equal to the mean demand λij of an ithconsumer item at each jth location; (d) determining a first locationthat meets a pre-selected allocation decision criterion comprising atleast the draw units allocated to each jth location for the i^(th)customer item, and the mean demand λ_(ij) for the i^(th) customer itemat each j^(th) location; (e) allocating a first one and only oneadditional draw unit to the first location; (f) determining a secondlocation that meets the pre-selected allocation decision criterion withthe first location having the first one and only one additional drawunit and allocating a second one and only one additional draw unit tothe second location; (g) repeating the step f above until at least oneconstraint is satisfied, the at least one constraint being based onperformance metrics including at least one of safety stock, expectedsellout, expected stockout, or expected return; and (h) communicatingthe distribution policy that indicates an optimal allocation of drawunits for each location for the single period inventory system; whereineach of steps d through h are performed by a software module executingon a computer.
 14. The method according to claim 13 wherein thepre-selected allocation decision criterion is a maximum incrementalavailability max_(i,j){F(λ_(ij),D_(ij)+1)−F(λ_(ij),D_(ij))} for consumeritems i=1, 2, . . . , m at locations j=1, 2, . . . , n subject to one ormore of the following constraints: ΣΣSS_(ij)≦Q, Σ_(j)SS_(ij)≦q₁ ^(i),Σ_(i)SS_(ij)≦q₂ ^(j), % EST(λ,D)≦s, % ER(λ,D)≦r, % ESO(λ,D)≦e,a_(ij)≦D_(ij)≦b_(ij); A≦ΣΣD_(ij)≦B, A₁ ^(j)≦Σ_(i)D_(ij)≦B₁ ^(j), and A₁^(i)≦Σ_(j)D_(ij)≦B₁ ^(i), where F is the cumulative probabilitydistribution function (cdf) of demand for the i^(th) consumer item to aj^(th) location, D_(ij) is the draw or delivered quantity of an i^(th)consumer item to a j^(th) location, SS_(ij) is the safety stock ordifference between the draw of an i^(th) consumer item at a j^(th)location and a demand forecast at that location, Q is a total safetystock threshold for delivery of all m consumer items to all n locations,q₁ ^(i) is a safety stock of an i^(th) consumer item at all locations,q₂ ^(j) is a safety stock of all consumer items at a j^(th) location,EST is expected stockout, ER is expected return, ESO is expectedsellout, s is a predetermined expected stockout percentage threshold, ris a predetermined expected return percentage threshold, e is apredetermined expected sellout percentage threshold, a_(ij) and b_(ij)are respectively lower and upper boundaries for a draw of an i^(th)consumer item at a j^(th) location, A₁ ^(j) and B₁ ^(j) are respectivelylower and upper boundaries for to draw of all m consumer items at aj^(th) location, A₁ ^(i) and B₁ ^(i) are respectively lower and upperboundaries for the draw of an i^(th) consumer item at all n locationsand A and B are respectively lower and upper boundaries for the draw ofall m consumer items at all n locations.
 15. The method according toclaim 13 wherein the pre-selected allocation decision criterion is aminimum availability min_(ij){F(λ_(ij),D_(ij))} for consumer items i=1,2, . . . , m at locations j=1, 2, . . . , n subject to one or more ofthe following constraints: ΣΣSS_(ij)≦Q, Σ_(j)SS_(ij)≦q₁ ^(i),Σ_(i)SS_(ij)≦q₂ ^(j), % EST(λ, D)≦s, % ER(λ, D)≦r, % ESO(λ, D)≦e,a_(ij)≦D_(ij)≦b_(ij); A≦ΣΣD_(ij)≦B, A₁ ^(j)≦Σ_(i)D_(ij)≦B₁ ^(j), A₁^(i)≦Σ_(j)D_(ij)≦B₁ ^(i), where F is to cumulative probabilitydistribution function (cdf) of demand for the i^(th) consumer item to aj^(th) location, D_(ij) the draw or delivered quantity of an i^(th)consumer item to a j^(th) location, SS_(ij) is the safety stock ordifference between to draw of an i^(th) consumer item at a j^(th)location and a demand forecast at that location, Q is a total safetystock threshold for delivery of all m consumer items to all n locations,q₁ ^(i) a safety stock of an i^(th) consumer item at all locations, q₂^(j) is a safety stock of all consumer items at a j^(th) location, ESTis expected stockout, ER is expected return, ESO is expected sellout, sis a predetermined expected stockout percentage threshold, r is apredetermined expected return percentage threshold, e is a predeterminedexpected sellout percentage threshold, a_(ij) and b_(ij) arerespectively lower and upper boundaries for a draw of an i^(th) consumeritem at a j^(th) location, A₁ ^(j) and B₁ ^(j) are respectively lowerand upper boundaries for the draw of all m consumer items at a j^(th)location, A₁ ^(i) and B₁ ^(i) are respectively lower and upperboundaries for the draw of an i^(th) consumer item at all n locationsand A and B are respectively lower and upper boundaries for the draw ofall m consumer items at all n locations.
 16. The method according toclaim 13 wherein the pre-selected allocation decision criterion is aminimum incremental expected returnmin_(ij){ER(λ_(ij),D_(ij)+1)−ER(λ_(ij),D_(ij))} for consumer items i=1,2, . . . , m at locations j=1, 2, . . . , n subject to one or more ofthe following constraints: ΣΣSS_(ij)≦Q, Σ_(j)SS_(ij)≦q₁ ^(i),Σ_(i)SS_(ij)≦q₂ ^(j), % EST(λ, D)≦s, % ER(λ, D)≦r, % ESO(λ, D)≦e,a_(ij)≦D_(ij)≦b_(ij); A≦ΣΣD_(ij)≦B, A₁ ^(j)≦Σ_(i)D_(ij)≦B₁ ^(j), and A₁^(i)≦Σ_(j)D_(ij)≦B₁ ^(i), where D_(ij) is the draw or delivered quantityof an i^(th) consumer item to a j^(th) location, SS_(ij) is the safetystock or difference between the draw of an i^(th) consumer item at aj^(th) location and a demand forecast at that location, Q is a totalsafety stock threshold for delivery of all m consumer items to all nlocations, q₁ ^(i) is a safety stock of an i^(th) consumer item at alllocations, q₂ ^(j) is a safety stock of all consumer items at a j^(th)location, EST is expected stockout, ER is expected return, ESO isexpected sellout, s is a predetermined expected stockout percentagethreshold, r is a predetermined expected return percentage threshold, eis a predetermined expected sellout percentage threshold, a_(ij) andb_(ij) are respectively lower and upper boundaries for a draw of ani^(th) consumer item at j^(th) location, A₁ ^(j) and B₁ ^(j) arerespectively lower and upper boundaries for the draw of all m consumeritems at a j^(th) location, A₁ ^(i) and B₁ ^(i) are respectively lowerand upper boundaries for the draw of an i^(th) consumer item at all nlocations and A and B are respectively lower and upper boundaries forthe draw of all m consumer items at all n locations.
 17. The methodaccording to claim 13 wherein the pre-selected allocation decisioncriterion is a maximum decremental expected stockoutmax_(ij){EST(λ_(ij),D_(ij))−EST(λ_(ij),D_(ij)1)} for consumer items i=1,2, . . . , m at locations j=1, 2, . . . , n subject to one or more ofthe following constraints: ΣΣSS_(ij)≦Q, Σ_(j)SS_(ij)≦q₁ ^(i),Σ_(i)SS_(ij)≦q₂ ^(j), % EST(λ,D)≦s, % ER(λ,D)≦r, % ESO(λ,D)≦e,a_(ij)≦D_(ij)≦b_(ij); A≦ΣΣD_(ij)≦B, A₁ ^(j)≦Σ_(i)D_(ij)≦B₁ ^(j), and A₁^(i)≦Σ_(j)D_(ij)≦B₁ ^(i), where D_(ij) is the draw or delivered quantityof an i^(th) consumer item to j^(th) location, SS_(ij) is the safetystock or difference between the draw of an i^(th) consumer item at aj^(th) location and a demand forecast at that location, Q is a totalsafety stock threshold for delivery of all m consumer items to all nlocations, q₁ ^(j) is a safety stock of an i^(th) consumer item at alllocations, q₂ ^(j) is a safety stock of all consumer items at a j^(th)location, EST is expected stockout, ER is expected return, ESO isexpected sellout, s is a predetermined expected stockout percentagethreshold, r is a predetermined expected return percentage threshold, eis a predetermined expected sellout percentage threshold, a_(ij) andb_(ij) are respectively lower and upper boundaries for a draw of anconsumer item at a j^(th) location, A₁ ^(j) and B₁ ^(j) are respectivelylower and upper boundaries for the draw of all m consumer items at aj^(th) location, A₁ ^(i) and B₁ ^(i) are respectively lower and upperboundaries for the draw of an i^(th) consumer item at all n locationsand A and B are respectively lower and upper boundaries for the draw ofall m consumer items at all n locations.
 18. The method according toclaim 13 wherein the pre-selected allocation decision criterion is givenby: w₁(% ER(λ, D)−% ER(λ, D⁰))+w₂(% ESO(λ, D⁰)−% ESO(λ, D)) subject toone or more of the following constraints: ΣΣSS_(ij)≦Q, Σ_(j)SS_(ij)≦q₁^(i), Σ_(i)SS_(ij)≦q₂ ^(j), % EST(λ, D)≦s, % ER(λ, D)≦r, % ESO(λ, D)≦e,a_(ij)≦D_(ij)≦b_(ij); A≦ΣΣD_(ij)≦B, A₁ ^(j)≦Σ_(i)D_(ij)≦B₁ ^(j), A₁^(i)≦Σ_(j)D_(ij)≦B₁ ^(i) and % ER(λ, D)=% ESO(λ, D), where w₁ and w₂ areweights, D⁰ is an initial draw matrix, D_(ij) is the draw or deliveredquantity of an i^(th) consumer item to a j^(th) location, SS_(ij) is thesafety stock or difference between the draw of an i^(th) consumer itemat a j^(th) location and a demand forecast at that location, Q is atotal safety stock threshold for delivery of all m consumer items to alln locations, q₁ ^(i) is a safety stock of an i^(th) consumer item at alllocations, q₂ ^(j) is a safety stock of all consumer items at a j^(th)location, EST is expected stockout, ER is expected return, ESO isexpected sellout, s is a predetermined expected stockout percentagethreshold, r is a predetermined expected return percentage threshold, eis a predetermined expected sellout percentage threshold, a_(ij) andb_(ij) are respectively lower and upper boundaries for a draw of ani^(th) consumer item at a j^(th) location, A₁ ^(j) and B₁ ^(j) arerespectively lower and upper boundaries for the draw of all m consumeritems at a j^(th) location, A₁ ^(i) and B₁ ^(i) are respectively lowerand upper boundaries for the draw of an i^(th) consumer item at all nlocations and A and B are respectively lower and upper boundaries forthe draw of all m consumer items at all n locations.
 19. The methodaccording to claim 13 wherein the pre-selected allocation decisioncriterion is given by: w₁(% ER(λ, D)−% ER(λ, D⁰))+w₂(% EST(λ, D⁰)−%EST(λ,D)) subject to one or more of the following constraints:ΣΣSS_(ij)≦Q, Σ_(j)SS_(ij)≦q₁ ^(i), Σ_(i)SS_(ij)≦q₂ ^(j), % EST(λ, D)≦s,% ER(λ, D)≦r, % ESO(λ, D)≦e, a_(ij)≦D_(ij)≦b_(ij); A≦ΣΣD_(ij)≦B, A₁^(j)≦Σ_(i)D_(ij)≦B₁ ^(j), A₁ ^(i)≦Σ_(j)D_(ij)≦B₁ ^(i) and % ER(λ, D)=%ESO(λ, D), where w₁ and w₂ are weights, D⁰ is an initial draw matrix,D_(ij) is the draw or delivered quantity of an i^(th) consumer item toj^(th) location, SS_(ij) is the safety stock or difference between thedraw of an i^(th) consumer item at a j^(th) location and a demandforecast at that location, Q is a total safety stock threshold fordelivery of all m consumer items to all n locations, q₁ ^(i) is a safetystock of an i^(th) consumer item at all locations, q₂ ^(j) is a safetystock of all consumer items at a j^(th) location, EST is expectedstockout, ER is expected return, ESO is expected sellout, s is apredetermined expected stockout percentage threshold, r is apredetermined expected return percentage threshold, e is a predeterminedexpected sellout percentage threshold, a_(ij) and b_(ij) arerespectively lower and upper boundaries for a draw of an i^(th) consumeritem at a j^(th) location, A₁ ^(j) and B₁ ^(j) are respectively lowerand upper boundaries for the draw of all m consumer items at a j^(th)location, A₁ ^(i) and B₁ ^(i) are respectively lower and upperboundaries for the draw of an i^(th) consumer item at all n locationsand A and B are respectively lower and upper boundaries for the draw ofall m consumer items at all n locations.
 20. The method according toclaim 13, wherein the pre-selected allocation decision criterion is amaximum incremental availability criterion and the determining the firstlocation that meets the pre-selected allocation decision criterioncomprises determining an incremental availability for each j^(th)location, selecting a location with the maximum incrementalavailability.
 21. The method according to claim 13 wherein the consumeritem is a printed media publication.
 22. A machine-readable storagemedium storing a sequence of instructions execution of which causes aprocessor to determine a distribution policy for a single periodinventory system, to control distribution of a consumer item to each ofa plurality of locations of the single period inventory system, theexecution of the sequence of instructions causes the processor toperform the actions of: forecasting a mean demand of the consumer itemat each j^(th) location of the plurality locations on the basis of ahistorical sales data for the consumer item at each j^(th) location ofthe plurality of locations, the sales data stored in a memory; andoptimizing allocation of the consumer item to each of the plurality oflocations, the optimization comprising the steps of: a) allocating toeach j^(th) location an initial number of units of the consumer itemequal to the mean demand of the consumer item at each j^(th) location;b) computing an incremental availability criterion for each j^(th)location, the criterion including at least the number of consumer itemsallocated to each j^(th) location and the mean demand λ_(ij) for thecustomer item at each j^(th) location; c) selecting a first locationwith the maximum incremental availability and allocating a first one andonly one additional number of the consumer item to the first location;d) updating the number of consumer items allocated to the first locationwith the first one and only one additional number of the consumer item;e) recomputing the incremental availability criterion for each j^(th)location and selecting a second location with the maximum incrementalavailability criterion and allocating a second one and only oneadditional number of the consumer item to the second location; and f)repeating the step f above until at least one constraint is satisfiedthe at least one constraint being based on performance metrics includingat least one of safety stock, expected sellout, expected stockout orexpected return.
 23. The machine-readable storage medium of claim 22wherein availability is a probability of completely satisfying a demandfor the consumer item at a j^(th) location without occurrence of asellout due to insufficient availability of the consumer item at thatlocation and is computed by a cumulative distribution function (cdf) ofthe mean demand λ_(1j) and draw D_(1j) of the consumer item allocatedfor that j^(th) location, wherein the incremental availability isdifference between a first availability of the consumer item at thatj^(th) location and a second availability of the consumer item at thatj^(th) location, the second availability computed with one additionaldraw unit D_(ij) of the consumer item allocated at that j^(th) locationin comparison to the draw units allocated for computation of the firstavailability and then determining from the result of the subtraction amaximum value associated with the consumer item at a location whereinthe determining the first location determines a location that has themaximum incremental availability.